Problem 41 - Understanding Elementary Algebra with Geometry

Question

Determine the slope of the line from its equation. $$x+y=7$$

Step-by-Step Solution

Answer

The slope of the line is \(-1\).

Step 1: Rewrite the Equation

Rewrite the given equation in the slope-intercept form, which is written as \[ y = mx + b \]here, \(m\) represents the slope and \(b\) represents the y-intercept. The given equation is \[ x + y = 7 \]First, solve for \(y\).

Step 2: Isolate the Variable y

Subtract \(x\) from both sides of the equation to isolate \(y\):\[ x + y - x = 7 - x \]which simplifies to \[ y = -x + 7 \]

Step 3: Identify the Slope

Now that the equation is in the slope-intercept form \[ y = -x + 7 \]identify the slope \(m\) as the coefficient of \(x\). In this case, the coefficient of \(x\) is \(-1\). So, the slope is \(-1\).

Key Concepts

slope-intercept form

When working with linear equations, it's incredibly useful to write them in the slope-intercept form. This form is expressed as \( y = mx + b \). Here, \( m \) stands for the slope of the line, and \( b \) represents the y-intercept, where the line crosses the y-axis. By rearranging equations into this form, it becomes easy to immediately see both the slope and y-intercept. For example, in the given problem, the equation \( x + y = 7 \) was rearranged to \( y = -x + 7 \) to identify the slope and y-intercept quickly.

linear equations

Linear equations represent relationships where the dependent variable changes at a constant rate with the independent variable. The general form of a linear equation in two variables is \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants. These equations graph as straight lines. To make working with them simpler, they can often be rewritten into the slope-intercept form. This transformation lets you easily spot the slope and the y-intercept. For instance, from the equation \( x + y = 7 \), we can rewrite it to \( y = -x + 7 \), making it clear that the slope of the line is \(-1\) and its y-intercept is \(7\).

solving for y

Solving for \( y \) in an equation is an important step in converting any linear equation to the slope-intercept form. It involves isolating \( y \) on one side of the equation. Let's look at the example provided. The equation \( x + y = 7 \) needs to be rearranged to solve for \( y \).

• Start by subtracting \( x \) from both sides: \( x + y - x = 7 - x \)
• This simplifies to \( y = -x + 7 \).

Now the equation is in the form \( y = mx + b \), making it straightforward to identify the slope and y-intercept. For this problem, the slope \( m \) is \(-1\) and the y-intercept \( b \) is \(7\).